Chapter 09 Introduction to the t Statistic
8. A confidence interval for estimating the mean An industrial/organizational psychologist wants to improve worker productivity for a client firm, but first he needs to gain a better understanding of the life of the typical white-collar professional.
"How many work-related emails do you receive during a typical workday?" For the subsample of legal professionals (n = 250), the mean response was M = 36.7 emails, with a sample standard deviation of s = 21.2 emails. - The estimated standard error is sMM = 1.341 .
She finds that the estimated Cohen's d is... , the t statistic is 2.38, and r² is... Using Cohen's d and Cohen's guidelines for interpreting the effect size with the estimated Cohen's d, there is a... treatment effect. Using r² and the extension of Cohen's guidelines for interpreting the effect size with r², there is a... treatment effect.
- 0.200 - 0.038 - small - small to medium
She finds that the estimated Cohen's d is ___ the t statistic is 3.64, and r² is ___ Using Cohen's d and Cohen's guidelines for interpreting the effect size with the estimated Cohen's d, there is a ___ treatment effect. Using r² and the extension of Cohen's guidelines for interpreting the effect size with r², there is a ____
- 0.333 - 0.099 - small to medium - medium
Now suppose you have a larger sample size n = 81. Calculate the estimated standard error and the t statistic for this sample with the same sample average and the same standard deviation as above, but with the larger sample size. The new estimated standard error is
- 0.5006 . - The new t statistic is1.80 .
One of the psychologist's findings is that six months after the workshop, a sample of 49 job seekers who received training on resume writing scored an average of 6.5 as measured on a 9-point job-search satisfaction scale, with a standard deviation of 1.2. The typical job seeker scores 5.8 points. The psychologist finds that the estimated Cohen's d is ... the t statistic is 4.12, and r² is...
- 0.583 - 0.261 .
Now suppose you have a larger sample size n = 89. Calculate the estimated standard error and the t statistic for this sample with the same sample average and the same standard deviation as above, but with the larger sample size. The new estimated standard error is __ . The new t statistic is ___
- 0.6679 - 1.35
The estimated standard error is ___.
- 1.0360 . - -2.32 .
In order to use the t distribution, you will first need to determine the degrees of freedom (df) for α = 0.05. The degrees of freedom (df) is 29 . The critical value of t is1.699 .
- 29 - 1.699
2. Comparing the t and the z statistics You can use both the t statistic and the z statistic to test hypotheses about the mean of population. The test that uses the t statistic is typically referred to as a t test, while the test that uses z statistic is commonly called a z test. Which of the following statements are true of the t statistic? Check all that apply.
- The formula for the t statistic is t = (M - μ) / sM - When you do not know the population standard deviation, you can use the t statistic, assuming all relevant assumptions are satisfied.
2. Comparing the t and the z statistics You can use both the t statistic and the z statistic to test hypotheses about the mean of population. The test that uses the t statistic is typically referred to as a t test, while the test that uses z statistic is commonly called a z test. Which of the following statements are true of the t statistic? Check all that apply.
- The t statistic could be considered as an estimated z statistic. - The t statistic provides a relatively poor estimate of z with small sample sizes.
Comparing the t and the z statistics You can use both the t statistic and the z statistic to test hypotheses about the mean of population. The test that uses the t statistic is typically referred to as a t test, while the test that uses z statistic is commonly called a z test. Which of the following statements are true of the t statistic? Check all that apply.
- The t statistic provides an excellent estimate of z, particularly with small sample sizes. - The t statistic uses the same formula as the z statistic except that the t statistic uses the estimated standard error in the denominator.
Therefore, the researcher ___ conclude that SAM-e has a significant effect on the moods of HIV patients.
- can
The t statistic ___ lie in the critical region. Therefore, you reject the null hypothesis.
- does not - cannot
The t statistic ___ in the critical region. Therefore, the null hypo thesis is ___
- does not lie - not rejected.
Now suppose you have a larger sample size n = 95. Calculate the estimated standard error and the t statistic for this sample with the same sample average and the same standard deviation as above, but with the larger sample size. The new estimated standard
- error is0.5849 - The new t statistic is2.05
The t statistic ___ in the critical region. Therefore, the null hypothesis ___ rejected.
- lies - is
Using Cohen's d and Cohen's guidelines for interpreting the effect size with the estimated Cohen's d, there is a.... to large treatment effect. Using r² and the extension of Cohen's guidelines for interpreting the effect size using r², there is a... treatment effect.
- medium - large
What is the probability of getting the t statistic or something more extreme for the sample size of n = 35? ___ . What is the probability of getting the t statistic or something more extreme for the sample size of n = 89? ___ .
- p =0.2000 - p =0.0902
Suppose that the researcher uses the same sample but decides to construct a 80% confidence interval of the mean of the population. Compared with the center of the 90% confidence interval, the center of the 80% confidence interval ____ . Compared to the 90% confidence interval, the 80% confidence interval is ____
- remains the same - narrower.
Suppose that the researcher uses the same sample but decides to construct a 95% confidence interval of the mean of the population. Compared with the center of the 85% confidence interval, the center of the 95% confidence interval ___ Compared to the 85% confidence interval, the 95% confidence interval ____
- remains the same - is wider
Using Cohen's d and Cohen's guidelines for interpreting the effect size with the estimated Cohen's d, there is a ____ treatment effect. Using r² and the extension of Cohen's guidelines for interpreting the effect size using r², there is a ___treatment effect.
- small to medium - medium
The t statistic is
-0.74 .
What is the probability of getting the t statistic or something more extreme for the sample size of n = 33? p =
0.1293 .
Your sample of 14-year-old children with antisocial tendencies has an average score of 9.65 with a standard deviation of 3.47. Calculate the t statistic. To do this, you will first have to calculate the estimated standard error. The estimated standard error is____
0.63 ; The t statistic is 1.11
Your sample of 10-year-old children with antisocial tendencies has an average score of 14.50 with a standard deviation of 2.95. Calculate the t statistic. To do this, you will first have to calculate the estimated standard error. The estimated standard error is
0.66 ; The t statistic is 2.35 .
Your sample of 16-year-old children with antisocial tendencies has an average score of 14.35 with a standard deviation of 4.63. Calculate the t statistic. To do this, you will first have to calculate the estimated standard error. The estimated standard error is
0.78 ; The t statistic is 1.60
The estimated standard error is
1.0845 .
The psychologist can be 99% confident that the interval from ___ includes the unknown population mean µ.
1.4 to 2.6
In order to use the t distribution, you will first need to determine the degrees of freedom (df) for α = 0.05. The degrees of freedom (df) is33 . The critical value of t is
1.691
In order to use the t distribution, you will first need to determine the degrees of freedom (df) for α = 0.05. The degrees of freedom (df) is19 . The critical value of t is1.729 .
19 ; 1.729
Use the following Distributions tool to develop a 90% confidence interval estimate of the mean number of hours legal professionals spend on online research during a typical workday. The psychologist can be 90% confident that the interval from .... includes the unknown population mean µ.
2.2 to 3.2
Assume that the depression scores among patients taking SAM-e are normally distributed. You will first need to determine the degrees of freedom. There are
24 degrees of freedom.
Assume that the depression scores among patients taking SAM-e are normally distributed. You will first need to determine the degrees of freedom. There are29 degrees of freedom.
29
The psychologist can be 95% confident that the interval from ___ includes the unknown population mean µ.
34.1 to 39.3
With another study, where you also plan on evaluating a mean using the t statistic, you have a sample of n = 56 that has an SS of 275. What is the variance for the sample? 75,625 16.58 2.24 5
5
For a sample of n = 36 that has a sample variance of 1,296, what is the estimated standard error for the sample? 37 6.09 6 36
6
3. Calculating the degrees of freedom, the sample variance, and the estimated standard error for evaluations using the t statistic You are planning to evaluate the mean of a single continuous variable from a study with a sample of n = 10 using the t statistic. What are the degrees of freedom for the sample?
9
Suppose you have a sample of 30 14-year-old children with antisocial tendencies and you are particularly interested in the emotion of surprise. The average 14-year-old has a score on the emotion recognition scale of 8.95. (The higher the score on this scale, the more strongly an emotion has to be displayed to be correctly identified. Therefore, higher scores indicate greater difficulty recognizing the emotion).
Assume that scores on the emotion recognition scale are normally distributed. You believe that children with antisocial tendencies will have a harder time recognizing the emotion of surprise (in other words, they will have higher scores on the emotion recognition test). What is your null hypothesis stated using symbols? - H₀: μantisocialantisocial ≤ μgeneral population general population What is your alternative hypothesis stated using symbols? - H₁: μantisocialantisocial > μgeneral populationgeneral population
6. Factors that influence a hypothesis test with the t statistic Suppose you conduct a hypothesis test about a population mean when the standard deviation is unknown by calculating a t statistic and determining whether to reject the null hypothesis.
Assume that your t statistic is a positive value and that you will reject the null hypothesis when t is very large. For each of the potential changes to the components of your study design or the results listed below, determine the change to the t statistic, and then select "Increases," "Decreases," or "Stays the same" from the dropdown menu in the right-hand column. A switch from using a two-tailed test to a one-tailed test - Stays the same An increase in the sample variance (s²) - Decreases An increase in the significance level (such as using α = .05 instead of α = .01) - Stays the same A decrease in the sample size (n) - Decreases
S-adenosyl methionine (SAM-e) is a naturally occurring compound in human cells that is thought to have an effect on depression symptoms. Suppose that a researcher is interested in testing SAM-e on patients who are struggling with Alzheimer's. She obtains a sample of n = 30 patients and asks each person to take the suggested dosage each day for 4 weeks.
At the end of the 4-week period, each individual takes the Beck Depression Inventory (BDI), which is a 21-item, multiple-choice self-report inventory for measuring the severity of depression. The scores from the sample produced a mean of M = 28.9 with a standard deviation of s = 5.94. In the general population of Alzheimer's patients, the standardized test is known to have a population mean of μ = 29.7. Because there are no previous studies using SAM-e with this population, the researcher doesn't know how it will affect these patients; therefore, she uses a two-tailed single-sample t test to test the hypothesis. From the following, select the correct null and alternative hypotheses for this study: H₀: μSAM-eSAM-e = 29.7; H₁: μSAM-eSAM-e ≠ 29.7
The average duration of labor from the first contraction to the birth of the baby in women over 35 who have not previously given birth and who did not use any pharmaceuticals is 16 hours. Suppose you have a sample of 33 women who exercise daily, and who have an average duration of labor of 16.9 hours and a sample variance of 20.3 hours. You want to test the hypothesis that women who exercise daily have a different duration of labor than all women.
Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is sMM=0.7843 . The t statistic is1.15 .
The average age at which adolescent girls reach their adult height is 16 years. Suppose you have a sample of 25 adolescent girls who are developmentally delayed, and who have an average age at which they reached their adult height of 17.2 years and a sample variance of 32.5 years. You want to test the hypothesis that adolescent girls who are developmentally delayed have a different age at which they reached their adult height than all adolescent girls.
Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is sMM=1.1402 . The t statistic is 1.05 .
8. A confidence interval for estimating the mean An industrial/organizational psychologist wants to improve worker productivity for a client firm, but first she needs to gain a better understanding of the life of the typical white-collar professional.
Fortunately, she has access to the 2008 Workplace Productivity Survey, commissioned by LexisNexis and prepared by WorldOne Research, which surveyed a sample of 650 white-collar professionals (250 legal professionals and 400 other professionals). One of the survey questions was, "During the average workday, how many hours do you spend attending meetings?" For the subsample of legal professionals (n = 250), the mean response was M = 2.0 hours, with a sample standard deviation of s = 3.6 hours. - The estimated standard error is sMM =0.228 .
Another one of the psychologist's findings is that a sample of 81 job seekers who received training on interview skills scored an average of 8.1 as measured on a 9-point job search motivation scale, with a standard deviation of 0.8. The typical job seeker scores 7.4 points.
He finds that the estimated Cohen's d is .875 , the t statistic is 7.78, and r² is 0.431 . Using Cohen's d and Cohen's guidelines for interpreting the effect size with the estimated Cohen's d, there is a large treatment effect. Using r² and the extension of Cohen's guidelines for interpreting the effect size with r², there is a large treatment effect.
Normally the psychologist will not know the value of the population mean. But consider the (unrealistic) scenario that a census of legal professionals is conducted. The census reveals that the population mean is μ = 4.0. How would the psychologist most likely react to the news?
The psychologist would be surprised that μ = 4.0, because that value is outside the confidence interval.
Normally the psychologist will not know the value of the population mean. But consider the (unrealistic) scenario that a census of legal professionals is conducted. The census reveals that the population mean is μ = 41.2. How would the psychologist most likely react to the news?
The psychologist would be surprised that μ = 41.2, because that value is outside the confidence interval.
Normally the psychologist will not know the value of the population mean. But consider the (unrealistic) scenario that a census of legal professionals is conducted. The census reveals that the population mean is μ = 2.1. How would the psychologist most likely react to the news?
The psychologist would not be surprised that μ = 2.1, because that value is inside the confidence interval.
At the same confidence level, ____ in the sample size would have a similar effect on the width of the confidence interval as changing the confidence level from 85% to 95% .
a decrease
At the same confidence level, _____ in the sample size would have a similar effect on the width of the confidence interval as changing the confidence level from 80% to 90% .
a decrease
At the same confidence level, ____ in the sample size would have a similar effect on the width of the confidence interval as changing the confidence level from 90% to 80% .
an increase
The critical t scores (the values that define the borders of the critical region)
are ± 2.756
Use the t distribution table to find the critical region for α = 0.01. The t Distribution: The critical t scores (the values that define the borders of the critical region)
are ± 2.756 .
Therefore, the researcher conclude that SAM-e has a significant effect on the moods of Alzheimer's patients.
cannot
The t statisticdoes lie in the critical region. Therefore, youcan reject the null hypothesis.
does ; can
The t statisticdoes not lie in the critical region. Therefore, youcannot reject the null hypothesis.
does not ; can not
With the exception of whether the population standard deviation is known, the necessary assumptions for hypothesis tests with the t statistic and with the z statistic are
essentially the same
With the exception of whether the population standard deviation is known, the necessary assumptions for hypothesis tests with the t statistic and with the z statistic are
essentially the same .
The t distribution is____ and more spread out with a smaller n. (Hint: To best see this, click the radio button in the tool with no vertical lines. Slowly move the Degrees of Freedom slider from the smallest value to the largest value, and observe how the shape of the distribution changes.)
flatter
The t distribution is ___ with a smaller n. (Hint: To best see this, click the radio button in the tool with no vertical lines. Slowly move the Degrees of Freedom slider from the smallest value to the largest value, and observe how the shape of the distribution changes.)
flatter and more spread out
Based on the results of this test, there is enough evidence to conclude that children with antisocial tendencies have greater difficulty recognizing sadness than do children without antisocial tendencies.
is
Based on the results of this test, there is not enough evidence to conclude that children with antisocial tendencies have greater difficulty recognizing disgust than do children without antisocial tendencies.
is not
Based on the results of this test, there is not enough evidence to conclude that children with antisocial tendencies have greater difficulty recognizing surprise than do children without antisocial tendencies.
is not
Compared to the 80% confidence interval, the 90% confidence interval
is wider
Using Cohen's d and Cohen's guidelines for interpreting the effect size with the estimated Cohen's d, there is a
large
treatment effect. Using r² and the extension of Cohen's guidelines for interpreting the effect size using r², there is a
large
Note that the t statistic becomes ___ as n becomes larger.
larger
Note that the t statistic becomes ____ as n becomes larger.
larger
The t distribution is ___ (Hint: To best see this, click the radio button in the tool with no vertical lines. Slowly move the Degrees of Freedom slider from the smallest value to the largest value, and observe how the shape of the distribution changes.)
more like the z distribution with a larger n.
This is a one- tailed test. Given what you know, you will evaluate this hypothesis using a t statistic.
one ; t
What is the probability of getting the t statistic or something more extreme for the sample size of n = 95?
p =0.0216 .
What is the probability of getting the t statistic or something more extreme for the sample size of n = 81?
p =0.0378 .
What is the probability of getting the t statistic or something more extreme for the sample size of n = 25?
p =0.1521
Suppose that the researcher uses the same sample but decides to construct a 90% confidence interval of the mean of the population. Compared with the center of the 80% confidence interval, the center of the 90% confidence interval
remains the same
Suppose you have a sample of 35 women who exercise daily, and who have an average duration of labor of 16.9 hours and a sample variance of 39.7 hours. You want to test the hypothesis that women who exercise daily have a different duration of labor than all women. Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is __ . The t statistic is ___
sMM=1.0650 0.85
Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is ____.
sMM=1.1402 The t statistic is 1.05 .
Note that the t statistic becomes ___ as n becomes smaller.
smaller
S-adenosyl methionine (SAM-e) is a naturally occurring compound in human cells that is thought to have an effect on depression symptoms. Suppose that a researcher is interested in testing SAM-e on patients who are struggling with HIV. He obtains a sample of n = 25 patients and asks each person to take the suggested dosage each day for 4 weeks. At the end of the 4-week period, each individual takes the Beck Depression Inventory (BDI),
which is a 21-item, multiple-choice self-report inventory for measuring the severity of depression. The scores from the sample produced a mean of M = 20.8 with a standard deviation of s = 5.18. In the general population of HIV patients, the standardized test is known to have a population mean of μ = 23.2. Because there are no previous studies using SAM-e with this population, the researcher doesn't know how it will affect these patients; therefore, he uses a two-tailed single-sample t test to test the hypothesis. From the following, select the correct null and alternative hypotheses for this study: - H₀: μSAM-eSAM-e = 23.2; H₁: μSAM-eSAM-e ≠ 23.2
The critical t scores (the values that define the borders of the critical region) are
±2.064 .
7. Effect size - Cohen's d and r squared An industrial/organizational psychologist has been consulting with a company that runs weekend job-seeking workshops for the unemployed. She collected data on several issues related to these workshops and, after conducting statistical tests, obtained statistically significant findings. She needs to find a way to evaluate effect size so that she can make recommendations to the company.
One of the psychologist's findings is that 18 months after the workshop, a sample of 36 job seekers who received training on setting career goals scored an average of 7.3 as measured on a 9-point job-search motivation scale, with a standard deviation of 1.5. The typical job seeker scores 6.1 points. The psychologist finds that the estimated Cohen's d is - 0.800 , the t statistic is 4.80, and r² is0.397 .
8. A confidence interval for estimating the mean An industrial/organizational psychologist wants to improve worker productivity for a client firm, but first she needs to gain a better understanding of the life of the typical white-collar professional. Fortunately, she has access to the 2008 Workplace Productivity Survey, commissioned by LexisNexis and prepared by WorldOne Research, which surveyed a sample of 650 white-collar professionals (250 legal professionals and 400 other professionals).
One of the survey questions was, "During the average workday, how many hours do you spend attending meetings?" For the subsample of legal professionals (n = 250), the mean response was M = 2.0 hours, with a sample standard deviation of s = 3.6 hours. - The estimated standard error is sMM =0.228
One of the survey questions was, "During the average workday, how many hours do you spend conducting online research?" For the subsample of legal professionals (n = 250), the mean response was M = 2.7 hours, with a sample standard deviation of s = 4.5 hours.
The estimated standard error is sMM =0.285 .
One of the psychologist's findings is that immediately after the workshop, a sample of 81 job seekers who received training on using the Internet to find job listings worked more than 30 hours per week an average of 8.7 months in the last year, with a standard deviation of 4.1. The typical job seeker works 7.4 months.
The psychologist finds that the estimated Cohen's d is0.317 , the t statistic is 2.83, and r² is 0.091 .
Suppose the mean of a population is μ = 36. A researcher (who does not know that μ = 36) selects a random sample of size n from this population. Then he constructs a 90% confidence interval of the population mean.
The true population mean and the researcher's 90% confidence interval of the population mean are shown in the following graph. Use the graph to answer the questions that follow.To construct the confidence interval, the quantity tsMM is subtracted from and added to the sample mean. In this case, - tsMM = 2.5
9. Properties of a confidence interval Suppose the mean of a population is μ = 61. A researcher (who does not know that μ = 61) selects a random sample of size n from this population. Then she constructs an 80% confidence interval of the population mean. The true population mean and the researcher's 80% confidence interval of the population mean are shown in the following graph. Use the graph to answer the questions that follow.
To construct the confidence interval, the quantity tsMM is subtracted from and added to the sample mean. - In this case, tsMM = 3
9. Properties of a confidence interval Suppose the mean of a population is μ = 51. A researcher (who does not know that μ = 51) selects a random sample of size n from this population. Then she constructs an 85% confidence interval of the population mean. The true population mean and the researcher's 85% confidence interval of the population mean are shown in the following graph. Use the graph to answer the questions that follow.
To construct the confidence interval, the quantity tsMM is subtracted from and added to the sample mean. In this case, tsMM = 4.0
Suppose you have a sample of 35 16-year-old children with antisocial tendencies and you are particularly interested in the emotion of disgust. The average 16-year-old has a score on the emotion recognition scale of 13.10. (The higher the score on this scale, the more strongly an emotion has to be displayed to be correctly identified. Therefore, higher scores indicate greater difficulty recognizing the emotion). Assume that scores on the emotion recognition scale are normally distributed.
You believe that children with antisocial tendencies will have a harder time recognizing the emotion of disgust What is your null hypothesis stated using symbols? - H₀: μantisocialantisocial ≤ μgeneral population general population What is your alternative hypothesis stated using symbols? - H₁: μantisocialantisocial > μgeneral population general population
Suppose you have a sample of 20 10-year-old children with antisocial tendencies and you are particularly interested in the emotion of sadness. The average 10-year-old has a score on the emotion recognition scale of 12.95. (The higher the score on this scale, the more strongly an emotion has to be displayed to be correctly identified. Therefore, higher scores indicate greater difficulty recognizing the emotion). Assume that scores on the emotion recognition scale are normally distributed.
You believe that children with antisocial tendencies will have a harder time recognizing the emotion of sadness (in other words, they will have higher scores on the emotion recognition test). What is your null hypothesis stated using symbols? -H₀: μantisocialantisocial ≤ μgeneral population What is your alternative hypothesis stated using symbols? -H₁: μantisocialantisocial > μgeneral populationgeneral population
